Operator-Adapted Reproducing Kernel Geometries for Structure-Preserving Representation Learning

Why can the same operator look distorted in one geometry and perfectly conservative in another?

A vector norm is not intrinsic to its coordinate entries: it depends on the inner product. In standard coordinates we use \(\|c\|_2^2=c^*c\); in a feature-space geometry we can instead use \(\|c\|_G^2=c^*Gc\). Thus, the statement that a transformation “preserves length” must always specify which length.

Requiring this to equal \(\|c\|_G^2\) for every coefficient vector \(c\) yields the fundamental equation \(M^*GM=G\). This is the bridge from an intuitive geometric question to a precise matrix problem.

Not every operator admits an adapted positive-definite metric

Changing the inner product cannot erase genuine exponential growth, decay, or a nontrivial Jordan block. In finite dimensions, an invertible matrix admits a positive-definite solution of \(M^*GM=G\) if and only if it is diagonalizable over \(\mathbb C\) and all of its eigenvalues lie on the unit circle. This criterion separates geometries that can be designed from operators for which exact isometry is impossible.

The equivalence is for the finite-dimensional setting studied here; it is not a blanket statement about arbitrary infinite-dimensional operators.

Fix the functions and the operator; leave the geometry free.

Choose linearly independent complex-valued functions \(\psi_1,\ldots,\psi_m\) on a nonempty set \(X\). Their span is a finite-dimensional function space. The prescribed linear operator \(T\) acts on coefficient vectors through a matrix \(M\).

The coefficients \(c\) describe a function, but they do not determine how its length is measured. That additional choice is encoded by a Hermitian positive-definite matrix \(G\).

Where does the inverse Gram metric come from?

For a fixed \(y\in X\), the kernel section \(k_G(\cdot,y)\) has coefficient vector \(G^{-1}\Psi(y)\). Using the inner-product convention in the manuscript,

This is the reproducing property. It makes the connection between a positive-definite matrix and a genuine function-space kernel exact—not merely an analogy with metric learning.

Norm preservation becomes a matrix identity.

Since \(Tf_c=f_{Mc}\), asking the operator to preserve the RKHS norm for every function is equivalent to asking that a quadratic form remain unchanged for every coefficient vector:

Theorem 2.2. This equation is necessary and sufficient for \(T\) to act isometrically on the constructed RKHS. It is a linear matrix constraint in the unknown entries of \(G\), together with the essential requirement \(G\succ0\).

Similarity to a unitary matrix gives an explicit solution.

Suppose \(M=SDS^{-1}\), where \(S\) is invertible and \(D\) is diagonal with unit-modulus entries. Then \(D^*D=I\). Choose

This \(G\) is positive definite, and direct substitution gives

Conversely, if \(M^*GM=G\), then \(U=G^{1/2}MG^{-1/2}\) satisfies \(U^*U=I\). Hence \(M\) is similar to a unitary matrix. These two directions explain the exact existence criterion.

Describe the full solution set, not just one metric

Section 02 constructed one solution when the operator is diagonalizable with unimodular eigenvalues. But that construction is not unique: changing the inner product within invariant eigenspaces can preserve exactly the same operator. The classification determines all such choices.

Every invariant metric has this form

Here \(S^{-*}=(S^{-1})^*\). Each positive-definite block \(H_j\) may be chosen freely, and every solution of \(M^*GM=G\) arises in this way. The metric from Section 02 is recovered by setting every \(H_j=I_{m_j}\).

Consequently, classification of invariant inner products gives a corresponding family of reproducing kernels for the same finite-dimensional function space. The feature functions \(\Psi\) are held fixed; what changes is their Hilbert-space geometry.

Repeated eigenvalues create extra geometric freedom

Consider the diagonal unitary operator below, with two equal eigenvalues and a third distinct eigenvalue:

Its invariant metrics in eigen-coordinates are exactly

The free variables are \(a,b,c\in\mathbb R\) and \(z\in\mathbb C\): five real parameters, agreeing with \(2^2+1^2=5\). If the three eigenvalues were distinct, the off-diagonal \(z\) would be forced to zero and the dimension would drop to three.

Exact structure preservation without a unique kernel

The classification gives a feasible family of RKHS geometries in which the prescribed operator is exactly norm-preserving. Within that family, one may subsequently investigate conditioning, interpretability, or compatibility with data without sacrificing the identity \(M^*GM=G\).

This is a mathematical design opportunity, not a claim that the paper proves a particular data-driven criterion will choose the best kernel or improve downstream accuracy.

Hold the operator fixed; change how lengths are measured.

The manuscript constructs a 20-dimensional operator \(M\) that is similar to a unitary matrix. Its eigenvalues lie on the unit circle, but its Euclidean singular values need not equal one. A positive-definite matrix \(G\) is constructed so that \(M^*GM=G\). Equivalently, the matrix below is unitary:

Similarity preserves the eigenvalues, not the singular values. The latter depend on the chosen inner product, which is exactly what this experiment tests.

120 samples, pairwise kernel values, one normalized discrepancy

With \(N=120\) sampled vectors \(z_1,\ldots,z_N\in\mathbb R^{20}\) and bandwidth \(\gamma=3\), the paper measures every pairwise change in similarity:

Thus \(\varepsilon_k=0\) means all sampled pairwise kernel values are preserved. The reported adapted value \(1.12\times10^{-15}\) is a finite-precision numerical confirmation of the exact invariance identity.